Bridge to Abstract Mathematics: Mathematical Proof and Structures

4 SETS Chapter 1

theorems of set theory, as suggested by the various types of evidence you

encounter.

Basic Definitions and Notation

The notion of set is a primitive, or undefined, term in mathematics, anal-

ogous to point and line in plane geometry. Therefore, our starting point,

rather than a formal definition, is an informal description of how the term

"set" is generally viewed in applications to undergraduate mathematics.

REMARK (^1) A set may be thought of as a well-defined collection of objects.
The objects in the set are called elements of the set.
The elements of a set may be any kinds of objects at all, ranging from,
most familiarly, numbers to names of people, varieties of flowers, or names
of states in the United States or provinces in Canada. A set may even have
other sets as some or all of its elements (see Exercise 9).
We will adopt the convention that capital letters A, B, X, Y, are used to
denote the names of sets, whereas lowercase a, b, x, y, denotes objects viewed
as possible elements of sets. Furthermore, the expression a E A (E is the
Greek letter "epsilon" in lower case) represents the statement "the object
a is an element of the set A," and x 4 X represents the assertion that the
object x is not an element of the set X. The convention about the use of
upper- and lowercase letters may occasionally be dispensed with in the text
when inconvenient (such as in an example in which an element of a given
set is itself a set). However, it is especially valuable and will be adhered to
in setting up proofs of theorems in later chapters.
One advantage of having an informal definition of the term set is that,
through it, we can introduce some other terminology related to sets. The
term element is one example, and the notion well-defined is another. The
latter term relates to the primary requirement for any such description:
Given an object, we must be able to determine whether or not the object lies

• in the described set. Here are two general methods of describing sets; as we
  will soon observe,
  method.

METHODS OF

well-definedness has a particular bearing on the second

DESCRIBING SETS

The roster method. We describe a set by listing the names of its elements,

separated by commas, with the full list enclosed in braces. Thus A =

(1,2,3,4) or B = (Massachusetts, Michigan, California) are sets consist-

ing of four and three elements, respectively, described by the roster method.

Note that 2 E A and Michigan E B, but 5 4 A and Ohio 4 B.